Specifically, is it required/customary to back up a "reopen" vote with a meta question[...]?
It is certainly not required, and it is customary only when you need discussion as a significant component of getting the question reopened.
In my experience, however, asking on meta for reopening of your own or somebody else's closed question, after casting your reopen vote, will definitely speed up the reopen process if the question is worth reopening.
This is more important for a site like English.SE where a huge number of questions can get closed each day and you need to draw user attention to a question that needs to be reopened.
On Interpersonal.SE, the review of your first 'leading' reopen vote by reopen-enabled members is so brisk and systematic that simply casting a reopen vote sets the question on the track to quick reopening, provided 4 users agree with you that the question was either closed without good enough reason or has since been sufficiently edited to solve the reason for closure. If it is a borderline case that needs discussion to convince users, then it is a very good case for starting a meta question.
Finally, meta questions are almost always the best option to get a question reopened if you are the OP. By which I mean to say that you need that meta post to not only convince members but also get crucial feedback, especially from the original close-voters, in order to know what you need to do to get your question reopened. This is not like just casting a reopen vote on somebody's closed question, which may or may not find 4 supporting votes for reopening. You have to actively pursue all possible means to correct major defects to get your question reopened.
It's a general point of etiquette followed on many Stack Exchange sites that users will not contradict the wisdom of 5 close voters unless the question was substantially edited to improve, or closed by a clear case of error. When an early Q of mine was closed on English.SE in April 2017, a senior member advised me that
it is much easier to close a question than to get it reopened, because once a Q is closed by 5 voters or a moderator, 5 other voters or another moderator will need very compelling reasons to vote to reopen.
In short it's a matter of site policy and etiquette, and not a matter of opinion: a closed Q is rarely reopened without significant corrective editing. Appropriate editing followed by substantial constructive discussion on meta is very useful towards convincing members to vote to reopen.
It is also my experience that close voting is exceptionally fair, constructive and consistent on many Stack Exchange websites: if you raise a meta question and demonstrate how your own or another OP's question has been edited and improved to clear the objections that got it closed in the first place, most of the close-voters will themself vote to reopen the question.